real analysis HW

HW4: MA 34100. DUE 2/8.

1. Consider Q equipped with the canonical metric. Let E = {p ∈ Q : 2 < p2 < 3}. Show that E is closed and bounded in Q. Is E compact? Is E open in Q? (justify!)

2. Construct a compact set of real numbers whose limit points form a countable set.

3. Let

A = {z ∈ C : ∃ a0, . . . , an integers and not all zero for which a0zn+a1zn−1+· · ·+an−1z+an = 0}. Prove that A is countable (elements of A are called algebraic numbers). Do there exist real numbers which are not algebraic? (justify!)

4. Is every point of every open set E ⊂ R2 a limit point of E? How about if E is closed?.

5. Suppose {A1, A2, . . .} ⊂ X where X is a metric space. Show that (i) if Bn = ∪ni=1Ai, then Bn = ∪ni=1Ai; (ii) if B = ∪

∞ i=1Ai, then B ⊃ ∪

∞ i=1Ai; (iii) find an example which

illustrates that the set inclusion in (ii) can be strict.

6. Let T be a collection of subsets of a set X which (i) contains ∅ and X; (ii) contains the intersection of any finite collection of sets in T ; (iii) contains the union of any collection of sets in T . The set T is said to be a topology for X. Define a topology for X = R.

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